Intro
Issue
In order to simulate sylviculture with TROLL we need to implement a new sylviculture module inside TROLL model code. A first litterature review was completed by an interview with Laurent Descroix of the Office Nationale des Forêts. We discovered that rotten trees were not random and seemed to depend both on tree species and diameter. This document presents modelling of relation between rotten trees and their species and diameter.
In fact we have two different questions:
- Predict if a tree will be probed as rotten (models M, see rotten 2)
- Predict how much of tree volume is rotten (models N, see rotten 3)
First all M model can be written as follow:
Table 1: Models summary.| m | model |
|---|---|
| \(M\) | \(Rotten_n \sim \mathcal{B}(inv_{logit}(\beta_0 + \beta_1*dbh_n))\) |
| \(M_{p}\) | \(Rotten_n \sim \mathcal{B}(inv_{logit}(\beta_0 + \beta_1*dbh_n + {\beta_2}_p))\) |
| \(M_{p,s}\) | \(Rotten_n \sim \mathcal{B}(inv_{logit}(\beta_0 + \beta_1*dbh_n + {\beta_2}_p + {\beta_3}_s))\) |
| \(M_{s,wsg,p}\) | \(Rotten_n \sim \mathcal{B}(inv_{logit}({\beta_0}_p + {\beta_1}_s + {\beta_2}*wsg_n + \beta_3*dbh_n))\) |
| \(M_{wsg,p}\) | \(Rotten_n \sim \mathcal{B}(inv_{logit}({\beta_0}_p + {\beta_2}*wsg_n + {\beta_3}*dbh_n\))) |
We tested models M detailed in following tabs to find the better trade-off between:
- Complexity (and number of parameters)
- Convergence
- Likelihood
Results are shown for each models in each model tabs and summarized in Conclusion tab.
Graph
Trees probbed rotten and estimated dbh.
\(M\)
Model
\(M\): \(Rotten_n \sim \mathcal{B}(inv_{logit}(\beta_0 + \beta_1*dbh_n))\)
Model Predictions.
Parameters markov chains. Light grey area represents warmup iterations.
Convergence
\(P_{rotten} = inv_{logit}(-2.655 + 0.279*dbh)\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(M\) glm
Model
data$dbh <- data$dbhest/dbhmax
m <- glm(probed_rotten ~ dbh, family = 'binomial', data)
print(m)
Call: glm(formula = probed_rotten ~ dbh, family = "binomial", data = data)
Coefficients:
(Intercept) dbh
-3.209 2.757
Degrees of Freedom: 3733 Total (i.e. Null); 3732 Residual
Null Deviance: 3380
Residual Deviance: 3330 AIC: 3334Plots
Plot 1
Plot 2
summary(predict(m)) Min. 1st Qu. Median Mean 3rd Qu. Max.
-2.2170 -1.8860 -1.6650 -1.6300 -1.4450 -0.4521 summary(data$probed_rotten) Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0000 0.0000 0.0000 0.1679 0.0000 1.0000 \(M_{p}\)
Model
\(M_{p}\): \(Rotten_n \sim \mathcal{B}(inv_{logit}(\beta_0 + \beta_1*dbh_n + {\beta_2}_p))\)
Model Predictions.
Parameters markov chains. Light grey area represents warmup iterations.
Convergence
\(P_{rotten} = inv_{logit}(-2.804 + 0.003*dbh)\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(M_{p,s}\)
Model
\(M_{p,s}\): \(Rotten_n \sim \mathcal{B}(inv_{logit}(\beta_0 + \beta_1*dbh_n + {\beta_2}_p + {\beta_3}_s))\)
Model Predictions.
Parameters markov chains. Light grey area represents warmup iterations.
Convergence
\(P_{rotten} = inv_{logit}(-2.989 + 0.008*dbh)\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(M_{s,wsg,p}\)
Model
\(M_{s,wsg,p}\): \(Rotten_n \sim \mathcal{B}(inv_{logit}({\beta_0}_p + {\beta_1}_s + {\beta_2}*wsg_n + \beta_3*dbh_n))\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(M_{wsg,p}\)
Model
\(M_{wsg,p}\): \(Rotten_n \sim \mathcal{B}(inv_{logit}({\beta_0}_p + {\beta_2}*wsg_n + {\beta_3}*dbh_n\)))
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
Conclusion
Model \(M_{s,p}0\) is the best one because it has the higher maximum likelihood. Moreover model \(M_{s,wsg,p}0\) shows us that adding \(\beta_2*wsg_n\) is not improving the model because \(\beta_2\) is centered on 0 and maximum likelihood stays the same. Consequently \(M_{wsg,p}0\) is not better than \(M_{s,p}0\).